Vector space

Definition

A vector space [ilmath]V[/ilmath] over a field [ilmath]F[/ilmath] is a non empty set [ilmath]V[/ilmath] and the binary operations:

• [math]+:V\times V\rightarrow V[/math] given by [math]+(x,y)=x+y[/math] - vector addition
• [math]\times:F\times V\rightarrow V[/math] given by [math]\times(\lambda,x)=\lambda x[/math] - scalar multiplication

Such that the following 8 "axioms of a vector space" hold

Axioms of a vector space

1. [math](x+y)+z=x+(y+z)\ \forall x,y,z\in V[/math]
2. [math]x+y=y+x\ \forall x,y\in V[/math]
3. [math]\exists e_a\in V\forall x\in V:x+e_a=x[/math] - this [math]e_a[/math] is denoted [math]0[/math] once proved unique.
4. [math]\forall x\in V\ \exists y\in V:x+y=e_a[/math] - this [math]y[/math] is denoted [math]-x[/math] once proved unique.
5. [math]\lambda(x+y)=\lambda x+\lambda y\ \forall\lambda\in F,\ x,y\in V[/math]
6. [math](\lambda+\mu)x = \lambda x+\mu x\ \forall\lambda,\mu\in F,\ x\in V[/math]
7. [math]\lambda(\mu x)=(\lambda\mu)x\ \forall\lambda,\mu\in F,\ x\in V[/math]
8. [math]\exists e_m\in F\forall x\in V:e_m x = x[/math] - this [math]e_m[/math] is denoted [math]1[/math] once proved unique.

Notation

We denote a vector space as "Let [math](V,F)[/math] be a vector space" often we will write simply "let [math]V[/math] be a vector space" if it is understood what the field is, because mathematicians are lazy

Example

Take [math]\mathbb{R}^n[/math], an entry [math]v\in\mathbb{R}^n[/math] may be denoted [math](v_1,...,v_n)=v[/math], scalar multiplication and addition are defined as follows:

• [math]\lambda\in\mathbb{R},v\in\mathbb{R}^n[/math] we define scalar multiplication [math]\lambda v=(\lambda v_1,...,\lambda v_n)[/math]
• [math]u,v\in\mathbb{R}^n[/math] - we define addition as [math]u+v=(u_1+v_1,...,u_n+v_n)[/math]

Important concepts

• Linear maps - the homomorphisms and isomorphisms of vector spaces
• Span, linear independence, linear dependence, basis and dimension
• Norm